In section 2.6 we discussed that ionospheric current is a function of magnetic field strength B. Richmond [57] and Cnossen et al. [12] found theoretically that ionospheric conductances are scaled with B. To our knowledge, these scaling factors have not been confirmed empirically, and we are not aware of any scaling factors confined to the ionosphere in darkness.
Magnetic perturbation data revealed a linear relationship betweenδBSH andδBN H. An example of this is given in Figure 4.9. The relation betweenδBSH and δBN H without the constant term can be expressed on the form
δBSH =b δBN H (5.2) where bis the slope of the linear equation which can be obtained by regression analysis.
Our datasets contains many outliers and hence motivates a more robust analysis than simple least squares. The slope b in equation (5.2) were therefore found by linear regression with Huber weights. Regression with Huber weights is an iterative scheme designed to reduce sensitivity to outliers (Huber [31]). The weights wi are defined as
wi = min
where k is set equal to 1.5 andhxii is the mean of the dataset. At first the linear model was obtained by linear least squares from the raw data. σ was calculated as the root mean square of the residuals. A new linear model was found with weighted regression of least squares using the weights found in equation 5.3. The Huber weights were calculated again using the new linear model andσ. This procedure was repeated until the deviation the new and old model was less than 0.001. A demonstration of the technique is shown in Figure 5.32. We see that the least square regression line is affected by the outliers while the regression line with Huber weights is not.
Figure 5.32: Simple linear regression (red line) is sensitive to outliers and gives a different regression than expected. Regression using Huber weights (green line) gives
model closer to the inliers
If the conjugate point between the hemispheres have identical equivalent currents we expect a slope equal to one in equation (5.2). Furthermore, if the current systems are dependent on B we would also suspect a slope near unity if the conjugate points had
equal magnetic field strengths. We tested this by comparing the ratio of the magnetic perturbations (δBSH/δBN H) to the magnetic field strengths (BSH/BN H). The former ratio can be determined from the slope found in equation (5.2) by rearranging the terms such thatb =δBSH/δBN H. The magnetic field strength at the location of the magnetometer stations were calculated using equation (4.6) from the IGRF model. These ratios were calculated for each station pair and the results can be found in Figure 5.33.
If a station in NH had more than one conjugate station, an average slope was calculated.
Numerical values of the slopes, correlation coefficients and number of data is shown in Table 5.2
Stations r r2 bqd bsm N
SCO-B10 0.577 0.332 1.265 0.998 42545 NAQ-B12 0.808 0.653 1.157 0.853 204050 FHB-B14 0.800 0.640 1.181 0.907 67881 GHB-B15 0.767 0.588 1.230 0.969 90568 GHB-B16 0.791 0.626 1.164 0.948 25569 GHB-B17 0.742 0.551 1.167 0.949 75977 GHB-B18 0.772 0.595 1.108 0.909 117175 SKT-B19 0.746 0.557 1.081 0.872 132601 IQA-B19 0.621 0.386 0.812 0.824 97077 STF-B19 0.656 0.431 1.029 0.862 106113 CY0-B20 0.727 0.528 0.786 0.842 4700 UPN-B20 0.659 0.434 0.609 0.641 37187 PGC-B21 0.500 0.250 0.655 0.524 11942 ATU-B21 0.680 0.462 0.965 0.812 73199 GDH-B22 0.671 0.450 0.763 0.659 103022 UMQ-B22 0.546 0.298 0.608 0.569 42304 IQA-B23 0.722 0.522 0.867 0.872 94158 HRN-DVS 0.686 0.470 0.689 0.734 136130 LYR-DVS 0.688 0.473 0.775 0.811 100314 JAN-MAW 0.786 0.618 1.011 0.815 329693 KOT-MCQ 0.846 0.716 0.764 0.745 34798 GDH-PG2 0.671 0.450 0.758 0.641 10183 UMQ-PG2 0.546 0.298 0.710 0.594 8772 PGC-SPA 0.585 0.342 0.738 0.800 52363
Table 5.2: Statistical information of each SuperMAG magnetometer pair. r is the correlation coefficient,N is the number of data point in each data set andbQD andbSM
are the slopes given in QD and SuperMAG coordinates respectively
Figure 5.33: Ratio of magnetic field strength compared to the strength of magnetic perturbations. The upper figure shows the ratios given in QD-coordinates while the
lower figure shows a equivalent plot given in SuperMAG-coordinates
Figure 5.34: Contours of QD-coordinates at 0km altitude. The black circles indicate where a majority of the magnetometer stations in our analysis are found.
Figure from Laundal and Richmond [42]
The QD-coordinate system is designed to remove effects from the local magnetic field.
In order to demonstrate this, consider an electrojet flowing strictly in QD-coordinates.
Such a current would produce constantNqdperturbations along the same QD longitude.
The current may flow over regions with varying local magnetic field and from equation 4.18 we can see that the QD base vectors acts to scale the Nqd component to be constant over varying local magnetic fields (Laundal and Gjerloev [41]). Other coordinate systems, such as the NEZ-coordinate system used by SuperMAG, does not compensate for features in the local magnetic field and magnetic perturbations would have a longitudinal dependences even though the current density remains constant. In Figure 5.33 we can see the differences in magnetic perturbations due to magnetic coordinate systems.
A large section of magnetometers in SH was found in a region with weaker magnetic field (black circle in the left map in Figure 5.34) compared to the magnetic field in the region of magnetometer stations found in NH (black circle in the left map in Figure 5.34). Due to the weaker magnetic field the current density measured in SuperMAG components would decrease, but QD-components compensate for this effect. This explains why the δBSH/δBN H have higher values in QD components than SuperMAG components, which is shown in Figure 5.33.
Even though QD-coordinates removes effects of features in the magnetic field, it does not remove conductivity effects. Ionospheric conductivity depends of the magnetic field strength both directly and by altering the gyration frequency. Conductance is roughly inversely proportional with magnetic field strength and ionospheric currents and also magnetic perturbations would be expected to be stronger at weaker magnetic fields. This is consistent with the result we found in Figure 5.33 and we therefore interpret this as conductivity effects.
We assume that magnetic perturbation measured on the ground is proportional to the strength of the above current systems which is dependent on the ionospheric conductances which is scaled with the magnetic field strength. We therefore tested whether the data in Figure 5.33 can be fitted on the model
δBSH
wherepis the scaling factor andAis a constant. Regression analysis gives the parameters
A= 0.79±0.04 (5.5)
p=−3.0±0.6 (5.6)
A plot of this result is shown in Figure 5.35.
Figure 5.35: Scaling factor with all available data
This result is much higher than the scaling factors reported by Richmond [57] and Cnossen et al. [12]. If our result is correct, then it would imply that the ionosphere in darkness has higher dependency of the magnetic field strength than the sunlit ionosphere.
However, the precision of the estimated parameters may be questioned. Some of our bqd=δBSH/δBN H data points were calculated from poorly correlated data sets. The slope is more precisely determined for correlated data.
Low correlations have effects on the slopes and the determination of the scaling factor p. It can be shown that the slope in a linear model is equivalent to b=rssy
x wherer is the correlation coefficient andsx and sy are the standard deviations of data set xandy respectively (Kennedy and Keeping [35]). Low correlation coefficients suppress the value of the slope bqd and a steeper curve must be fitted to the dataset. This give a higher scaling factor than expected. Therefore, we used only high correlated data sets for a more accurate determination ofbqd and the scaling factor p. If we required at least half of our data to be explained by linear model, i.e. r2 ≥0.5, then regression analysis of equation (5.4) gives the parameters
A= 0.89±0.04 (5.7)
p=−1.9±0.5 (5.8)
Equation (5.4) with parameters A and pgiven by equations (5.8) and (5.8) is shown in Figure 5.36.
Figure 5.36: Plots of equation 5.4 with parameter from our result (blue line) Cnossen et al. [12] (green line) and Richmond [57] (red line). The blue dots are slope from our
calculations.
We find that the scaling factor within one standard deviation is equal top= [−2.4,−1.4].
The scaling factor reported by Cnossen et al. [12] is within our results but the scaling factor reported by Richmond [57] is outside. Therefore, we agree with the result reported by Cnossen et al. [12]. However, the model in equation in Figure 5.4 were fitted from only 8 data points. The reliability of this result is therefore questionable. In the calculations of the slope we assumed that the conductances and their corresponding magnetic perturbations were dependent on the strength of the magnetic field, but it depends on other variables as well such as collision frequency. A better method could be using correlation coefficients as weights instead of rejecting data points.
Summary and Conclusion
We found theoretically that the ionosphere is in darkness for solar zenith angles greater than 100◦. However, measurements of magnetic perturbations at magnetic conjugate points suggest that solar zenith angles as low as 80◦ is low enough to be considered dark.
The theoretical value reduced our dataset and we found no advantages in using this value.
Therefore, we used solar zenith angles defined by magnetic perturbation measurements.
Many inter-hemispherical asymmetries have been linked to IMF By which we have scrutinized using magnetic perturbation data. IMF By can cause longitudinal shift of magnetic foot points which optimize magnetic conjugacy. We examined whether this improves correspondence of equivalent currents at conjugate points, but we found the opposite effect. A possible explanation is that other IMFBy effects such as asymmetric convection and current systems are much more significant.
Theoretical studies suggest that ionospheric conductance is scaled with the strength of the magnetic field, but to our knowledge this has not been tested empirically. Our results were consistent with this and we were able to derive a scaling factor for the night side ionospheric conductance. However, we recommend some caution considering this result.
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Future Work
The statistical significance our analysis can be improved by including more data. This can be done by identify more magnetometer stations at conjugate points or by updating the data in already identified magnetometer stations. Some of our dataset were provided quite recent stations. For example, stations PG1, PG2 and PG3 covered the year 2013 only. For the year to come more data will be provided by SuperMAG and the analysis will be more reliable.
The model fit in section 5.3 should be weighted for a more robust analysis. The determination of the scaling factor were based on regression analysis of data points found by the slope of linear models. The accuracy of calculation of the slopes are dependent on the correlation of the data sets. Therefore, the correlation coefficients could be used as weights in the model fit.
We have treated the magnetic perturbations as a function of the magnetic field strength, because the ionospheric conductances are scaled with the this variable. However, iono-spheric conductivity (and conductance) is dependent on both magnetic field and the collision frequency. Therefore, we suggest using multi-variable analysis including both magnetic field strength and collision frequency. By using reasonable assumptions of the altitude of the ionospheric currents, temperature and the ion and electron composition we can estimate the collision frequency by the equation published by Lathuillere and Wickwar [40] or the International Reference Ionosphere (Bilitza et al. [5])
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In our analysis we assumed that all magnetic perturbation were generated by currents flowing in the ionosphere. However, under certain conditions a significant portion of the magnetic perturbations measured by ground based magnetometers might originate from currents induced from the ground (Tanskanen et al. [64], Viljanen et al. [72]). The ground based magnetometers are placed on different surfaces, such as land, coast and ice, and the surface conductivity might influence the magnetic perturbation measurements.
One should therefore separate ionospheric and surface currents, for example by using techniques developed by Pulkkinen et al. [54].
SuperMAG stations
Table A.1: List of conjugate SuperMAG stations. The columns Start and End tell when the station pairs where operative simultaneously. Date format is given in yyyy-mm-dd.
Hours tells when how many hours they were operative simultaneously 77
IAGA MLAT MLON GLAT GLON
Table A.2: Stations in NH. MLAT and MLON are latitude and longitude in magnetic coordinates and GLAT and GLON are latitude and longitude in geographic coordinates.
Coordiantes are given in degrees
Table A.3: Stations in SH. MLAT and MLON are latitude and longitude in magnetic coordinates and GLAT and GLON are latitude and longitude in geographic coordinates.
Coordiantes are given in degrees